HyBrSim { A Modeling And Simulation Environment For
HyBrSim – A Modeling and SimulationEnvironment for Hybrid Bond GraphsPieter J. Mosterman Institute of Robotics and MechatronicsDLR OberpfaffenhofenD-82230 WesslingPieter.J.Mosterman@dlr.deDecember 1, 2000Keywordshybrid systems, physical system modeling, bond graphs, hybrid bond graphs,hydraulic actuatorsAbstractBond graphs are a powerful formalism to model continuous dynamics of physical systems. Hybrid bond graphs introduce an ideal switching element, the controlled junction, to approximate continuous behavior that is too complex for numerical analysis (e.g., because of nonlinearities or steep gradients). HyBrSim isa tool for hybrid bond graph modeling and simulation implemented in Java anddocumented in this paper. It performs event detection and location based on a bisectional search, handles run-time causality changes, including derivative causality,performs physically consistent (re-)initialization, and supports two types of eventiteration because of dynamic coupling. It exports hybrid bond graph models inJava and C/C code that includes discontinuities as switched equations (i.e.,pre-enumeration is not required).1IntroductionPhysical systems can be modeled by sets of ordinary differential equations (ODEs) possibly supplemented by algebraic constraints (DAEs). Often these are composed fromlocal constituent equations of primitive elements and constraints imposed by a networkstructure that connects them [1, 2]. Pieter J. Mosterman is supported by a grant from the DFG Schwerpunktprogramm KONDISK.1
1.1Stiff SystemsContinuous physical system models may contain nonlinearities and steep gradients thatcomplicate numerical simulation. Integration methods such as the Backward DifferenceFormula (BDF) [3, 4] address these problems by reducing the simulation step-size, atthe cost of increased computational complexity.To illustrate, consider the hydraulic cylinder in Fig 1. To move the piston withinertia, mp , the intake valve with fluid inertia, Iin , and flow resistance, Rin , is adjustedto control oil-flow into the cylinder. The elasticity, Coil , and viscosity, Roil , of the oilin the chamber generate a pressure that opens a relief valve with inertia and resistance,Irel and Rrel , respectively, when the pressure exceeds a predefined safety threshold.pinIinRinRoilpsmpIrel RrelCoilmpvpFigure 1: Hydraulic cylinder with relief valve.Initially, the relief valve may be closed and when the intake valve closes completely,there is no flow of oil into the chamber, causing the piston velocity to quickly become 0.In a continuous model, a small leakage flow of the valve interacts with the fast dynamics induced by Coil and Roil , which leads to high frequency low amplitude oscillationsrequiring a small simulation step-size for a considerable time interval.For the particular problem, efficient variable step-size numerical integration may notbe possible or such solvers may not be available or suitable for the task at hand (e.g.,real-time simulation).1.2DiscontinuitiesStiff systems because of steep gradients can be simplified by abstracting the fast continuous transients into discontinuous changes and the models become of a mixed continuous/discrete, hybrid, nature [5]. Hybrid systems typically operate in piecewise continuous modes, modeled by ODEs and DAEs. Mode changes are most conveniently modeled2
by activating and deactivating constituent equations of model components which maychange causality. For example, when the intake valve in Fig. 1 closes, its constituentequation changes from enforcing a 0 pressure drop across it to enforcing 0 flow. Thesecausal changes may result in dynamic (i.e., run-time) changes in the state vector.Two situations can be classified; state variables become dependent on exogeneous variables, and state variables become algebraically related.In Fig. 1, when the intake valve closes and the relief valve is closed as well, if the oilelasticity and viscosity are abstracted away, the piston velocity is forced to 0, and, therefore, its momentum is no longer a state variable (first issue). In this mode, the requiredpressure build-up in the cylinder may be such that the relief valve opens. Now, the statevariables that correspond to Irel and mp , i.e., the fluid momentum prel and the momentum pp , become dependent (algebraically related) and the initial piston momentum hasto be distributed so that their values are mutually consistent (second issue).The closing of the intake valve and the opening of the relief valve follow each otherwith no continuous, differential equation, behavior in between. In general, (de)activatingblocks of equations may result in a sequence of consecutive mode changes that has toconverge before continuous behavior resumes [5].1.3HyBrSimHyBrSim (Hybrid Bond gRaph Simulator) is a modeling and simulation environmentto handle hybrid behaviors, implemented in Java. Instead of generating a global systemof equations, HyBrSim attempts to propagate known variable values (input and state)through the model topology at each evaluation, i.e., it is interpretive. The advantageof the interpretive approach lies in the flexible treatment of variable structure, i.e.,hybrid, models for which HyBrSim is developed specifically [6], It does not focus onsophisticated handling of pure continuous and discrete behaviors.Section 2 reviews the bond graph modeling and simulation approach and identifies thesupport provided by HyBrSim. Section 3 discusses the hybrid bond graph modelingapproach, general hybrid dynamic systems effects, and how these are facilitated byHyBrSim. Section 4 describes the export filter to C/C and Java and Section 5presents conclusions and future work.2Bond Graph Modeling and SimulationIn HyBrSim, dynamic behavior of physical systems is modeled by bond graphs [7, 8].3
Table 1: Bond graph rtiesresistance, Rcapacitance, Cinitial value, q0inertia, Iinitial value, p0amplitude, Eamplitude, F01TFGYtransformationratio, ntransformationratio, rrelatione Rff C Rdedte C1 f dt qC0e I dfdtRf I1 edt pI0e Ef Fei e 0Pi fi 0f i f0Pi ei 0ein neoutfin nfouteout rfinein rfoutBond Graph Modeling With HyBrSimBond graphs model the exchange of energy, power, between idealized physical processes,which allows for multi-domain modeling (e.g., electrical, mechanical, hydraulic).2.1.1The Power DomainEach power connection, bond, contains two conjugate variables, effort (e) and flow (f ),the product of which constitutes power, that correspond to an intensive variable (e.g.,pressure and voltage) and rate of change of an extensive variable (e.g., volume andcharge), respectively [9]. There are nine primitive bond graph elements, listed in Table 1,that represent lumped ideal behavior and exchange energy through ports.Irreversible Processes The ideal irreversible processes, R, dissipate energy and generate entropy. For example, in Fig. 1 the intake valve is modeled to have a dissipativeeffect, Rin . If the generated entropy does not affect dynamic behavior, this port (atpresent not supported by HyBrSim) is not shown.Reversible Processes Ideal reversible processes store energy without dissipation.These are marked C and I to indicate stored flow and stored effort, respectively, with initial values generalized displacement, q0 ( Ce0 ) and generalized momentum p0 ( If0 ).For example, in Fig. 1 the oil elasticity, Coil , stores oil compression. These variablescapture the state of the system. In general, one storage element can communicate energy to many different domains, however, at present, HyBrSim only supports one port4
Figure 2: Bond graph of cylinder with PID controller without the valves being modeled.storage.The Context The dynamics of the system environment are not modeled. Instead,interaction is modeled by ideal sources that can be of effort and flow type, i.e., Se andSf, respectively. In Fig. 1, the interaction is by the hydraulic pressure, pin and sumppressure, psmp , both of Se type.The Junction Structure A junction structure that consists of normal and weightedelements distributes power. The normal elements are of common effort type (e.g., Kirchoff’s current law), 0-junctions, or common flow type (e.g., Kirchoff’s voltage law),1-junctions. For example, in Fig. 1 the oil pressure in the chamber is the same on theintake path, relief path, and the piston, and can therefore be modeled by a 0-junction.The weighted distribution elements are of TF, the transformer, and GY, the gyratortype. The intake valve in Fig. 1 can be modeled by a TF element to modulate thehydraulic power supplied to the cylinder.Graphical Appearance A HyBrSim bond graph model of the cylinder in Fig. 1 isshown in Fig. 2. The bond graph elements are rectangles with the element type on theleft and their name on the right [10]. Power bonds (depicted by a harpoon) connectelements that exchange energy, e.g., in Fig. 2 the connection between the Se elementand the TF element. In addition, bond graph models may contain modulation effectsbased on connections that carry no energy. Such a connection may correspond to anindividual effort or flow variable that is tapped from a power bond by an active bond(depicted by an arrow). In Fig. 2 an active bond taps the velocity of the piston mp andfeeds its value into the displacement element that is part of the signal domain.5
Table 2: Block diagram elements.2.1.2elementclockIDtpropertiesp, x0integratorstepint-p, x0p, x0summultiplierinverter invpippsquarerootsincossqrtpsincospprelationif time p then so x0else s0 time x0 pRPso pi si dtPif i si p then so 1else s0 x0Pso i pi sis o p Π i si1so p Psi iso p p Pi si Pso p sin( i si )Pso p cos( i si )The Signal DomainA number of mathematical operations are available that operate on signals, i.e., connections that carry no power (of which active bonds are a subset). In HyBrSim, the signalpart of a model is shown in blue (gray in monochrome depiction).In Fig. 2 the signal part is used to model a PID control law for the piston displacement. This is a geometric state that is used as input to the control part, along with adesired setpoint. The error between the two is integrated once to implement integralcontrol and the velocity is used directly to add a derivative component. A set of proportional, integral, and derivative gains (Kp , Ki , Kd ) computes the controller outputused to modulate the physical process, in this case the hydraulic power supplied to thecylinder. These block diagram elements allow multiple input, si , and output, so , signals.Their functionality is given in Table 2.2.1.3User InterfaceThe HyBrSim user interface consists of a workspace and a bond graph and block diagram toolbox. Connections are made by first clicking the source and then the destinationelement. The source element determines the type of connection. If it is a bond graphelement, a power connection is made. If it is a block diagram element, a signal connection is made. A connection from a block diagram to a bond graph element is allowedin case of a modulated element, Se, Sf, TF, and GY, and automatically interpreted assuch.Right clicking on an element brings up a pop-up menu with entries that include theelement name and its properties. In most cases, element properties include a parameter,p, and initial value, x0 (Table 1 and Table 2). For the summing element it contains a6
drop down list with all ingoing signals and their sign, pi , i.e., whether they are to beadded or subtracted.2.2Simulation with HyBrSimContinuous simulation applies a distributed token passing approach and is based on theForward Euler method, x(tk 1 ) x(tk ) ẋ(tk ) T , where x(tk ) is the state at time tk ,ẋ(tk ) its time derivative, and T the integration step-size. Each port of a bond graphand block diagram element has an attribute that contains the current value of the tokenat that port [11].2.2.1The MethodFirst, causality is assigned to the power elements based on a sequential causality assignment procedure (SCAP) like algorithm [12]. Next, an execution order is determined suchthat, at a given evaluation, k, all input values of an element are known when it is calledto compute its output. Modulated elements are handled by introducing pseudo statebehavior, i.e., the modulation factor is one evaluation (during continuous integrationthis equals one integration step) delayed. The initial value is user specified.The execution order is determined by first propagating the values of all clock elementsso time modulation is not delayed. Next, the values of sources and storage elements inintegral causality are propagated. The values of sources are user provided or, in case ofmodulation, given by the pseudo state value. The values of storage elements in integralcausality are computed from their state. Storage elements in derivative causality areno propagation roots and compute a numerical difference approximation of the timek 1derivative variables, p1 xk x, where p is the parameter and xk the state with x0 as its Tinitial value. Note that this implies the first time step is off. Furthermore, zero-ordercausal paths between resistances cannot be handled. These issues are addressed by arecently implemented approach that is beyond the scope of this paper.After all effort and flow variables have assigned values, these are propagated into theblock diagram model part via active bonds. Along with the integrator elements thatpropagate their stored value and the values of clock elements, all block diagram variablevalues are computed.2.2.2Derivative CausalityIn case of derivative causality, there is dependency between storage elements and sourcesand straightforward propagation of their values does not apply. Instead, algebraic constraints on state variables determine their values. In case of dependency on sources only,the stored energy can be computed directly. If dependency on other storage elements7
Figure 3: Bond graph with derivative causality.exists, state values have to be found that are consistent with the algebraic constraints.Two critical issues must be solved: (i) a given set of values has to be made consistentwith the algebraic constraints, and (ii) it must be ensured that one integration step ofdynamic behavior remains consistent with the algebraic constraints.To achieve (i), HyBrSim implements the conservation of state principle [13] as aniteration process between algebraically related storage elements. In Fig. 3, I2 is inderivative causality1 because of the algebraic dependency between I1 and I2 throughtheir common flow. So the stored momentum p1 and p2 has to be consistent withp1p2 .I1I2(1)Suppose that at a point in time, tk , p1 2 and p2 1. These values may be userspecified if tk is the simulation start time or the result of a mode switch at tk in case of ahybrid model. For I1 I2 1, this is inconsistent with Eq. (1) and iteration first takespart of the stored momentum out of I1 , determined by a convergence factor, η 0.4,and transfers it into I2 . This leads to p1 1.6 and p2 1.4, see Table 3. For thesemomentum values, Eq. (1) is still not satisfied, and again a, now smaller, part of storedmomentum is transferred from I1 to I2 . This results in p1 1.52 and p2 1.48. Thisprocess continues till f1 f2 , within some preset numerical margin.An important observation is that this allows all storage elements to be initialized.Other simulation tools typically allow only initialization of the reduced state, which maylead to differences in the simulation results. For example, if the I elements in Fig. 3 arereplaced by C elements, standard simulation packages allow the initialization of onlyone of them and depending on the value of E, the state of the other is computed whileneglecting to take into account the constraint that only displacement can be added toone if it is taken out of the other.Once consistent values are found, it has to be ensured that future behavior remainsconsistent, i.e., for the model in Fig. 3 that Eq. (1) remains satisfied. In an equationbased system, this is achieved by computing the gradient of behavior while accounting1Note that the effort causality is indicated by a perpendicular stroke at the end of a power bondand flow causality by a circle at the other end.8
Table 3: Conservation of state iteration, η 0.4.iteration01234p121.61.521.5041.5008p 221.61.521.5041.5008p211.41.481.4961.4992η(p 2 p2 )0.40.080.0160.00320.00064for the dependency. From Eq. (1) andṗ1 E ṗ2 p1RI1it follows, after differentiation of Eq. (1), that ṗ1 1(I EI1 I2 1(2) Rp1 ) 3.25 and one T 0.1 time step gives p1 (tk 1 ) p1 (tk ) ṗ1 (tk ) T 1.175.HyBrSim takes a two step approach. First, a gradient is allowed that results instate values that are inconsistent with the algebraic constraints. Second, the iterationapproach is applied to find consistent values again. For the two inertias, after consistentvalues p1 p2 1.5 are found, the gradient of p1 is determined while disregarding theinfluence of I2 , i.e., e2 0, which yields e1 6.5, and one simulation step T 0.1 istaken that only updates the state of I1 , p1 0.85. This results in values for p1 and p2that violate the algebraic constraint in Eq (1) and iteration computes p1 p2 1.175as well.2.2.3User InterfaceEffort, flow, and signal values can be graphed after simulation. These variables arecoded by coloring the effort stroke, the flow circle, and the fill color of the signal arrowhead, corresponding to the colors of the traces in a graph. This graph has a small set ofrudimentary display features such as data points on/off, autoscale, user selected maximaand minima for x and y-axis separately, and a mouse driven zooming feature. To allowthe use of sophisticated plotting features such as those provided by Matlab [14], datacan be written to a file in standard ASCII format, which includes the evaluation step,k, and the time stamp, tk , for each of the sets of data points.2In addition, power along each bond can be animated (logarithmic or linear) withpositive power based at the harpoon destination and negative power at the source.Figure 4 shows the power distribution in the hydraulic actuator in Fig. 1 at t 0.2 s.The relief valve is considered to be closed, and, therefore, not shown and the PID controlpart is not shown because it does not distribute power. After some initial transient,2All plots in this paper are generated by Matlab.9
Figure 4: Logarithmic power distribution at t 0.2 s.the oil dynamics reach an internal steady state, and the parameters Roil and Coil donot consume any more power. Therefore, to investigate low frequency behavior theseelements can be removed, e.g., by a singular perturbation related approach for bondgraphs [15]. In general, such power analyses can be used to aid in model reductionby removing elements with low power consumption [16]. Animation can be pauzed,restarted, and continued from final values when the simulation end time was reached.3Hybrid Bond GraphsPiecewise linearization of nonlinearities and removal of steep gradients may lead to hybrid models with continuous behavior that is interspersed with discrete mode changes [17].3.1Hybrid Bond Graph ModelingHybrid bond graphs endow the bond graph modeling formalism with a finite state machinemodel part that communicates by means of a controlled junction [18].3.1.1From Discrete to ContinuousThe controlled junction operates in one of two states to systematically model switchingbehavior. When it is on, it acts as a normal junction, and when it is off a 0-junctionacts as a 0 value effort source and a 1-junction as a 0 value flow source (see Fig. 5).This implements ideal switching behavior (e.g., [19, 18, 20]) and changes causality onone port when the junction changes its state. Note that dissipation may still occur whenjunctions change their state, e.g., in case of a perfectly nonelastic collision.10
OFFONSe:0OFFONSf:001(b) 1-junction.(a) 0-junction.Figure 5: The controlled junction types.Figure 6: Hybrid bond graph model of actuator with the valves modeled by controlledjunctions.In the bond graph model in Fig. 2, if the valves are modeled as ideal switches, thecorresponding 1-junctions (in and relief) become controlled junctions with inertial anddissipative effects explicitly modeled.3.1.2The Discrete Event ModelThe discrete event model part is implemented by local finite state machines (FSM), onefor each controlled junction, that map each of their states onto the on and off states. Thegraphical representation is a state transition diagram that is associated to the junctionproperty. For example, Fig. 6 shows the FSM for the relief valve mechanism in Fig. 1.When the net pressure, pnet , crosses a threshold value, pth , pnet pcyl pth 0.0 therelief valve opens, i.e., the corresponding controlled junction comes on. Note that thethreshold value can also be modeled in the FSM. For example, the relief valve closesagain when the pressure has subsided and crosses another threshold (pnet 25.0).This leads to the behavior in Fig. 7: During a control maneuver, the intake valvecloses inadvertently at t 0.2 s. Shortly thereafter, the oil parameters have built up apressure in the cylinder chamber that exceeds the safety threshold. Consequently, the11
relief valve opens for a short duration, see Fig. 7(b), till the pressure has subsided andthen closes again. This leads again to too high a pressure in the cylinder and the sameprocedure repeats, after which the piston velocity has reduced to a value that can safelydecay to 0. This ‘stuttering’ is typical for relief valve behavior.Each FSM associated with a controlled junction has an initial state, indicated by ashaded background, and an active state that is highlighted. States can be added andremoved but are always of the on and off type.Because FSM switching effects are included locally, no global analysis of the modesof behavior is required. Though this avoids pre-enumeration (which can be prohibitivelycomplex because of the state explosion), it requires run-time facilities to determine thenew global mode dynamically. This includes causal analysis which may lead to run-timechanges in the complexity of the underlying system of equations (i.e., derivative causalitymay emerge).3.1.3From Continuous to DiscreteBlock diagram signals and active bonds that may cause a controlled junction to changeits state are connected to this junction and show up in the state transition diagram assignal ports. Signal ports constitute crossing functions that compare the value of thecorresponding variable in the bond graph, x, with a threshold, δ. Two inquiries areallowed: (i) the new truth value of the crossing function can be requested and (ii) itcan be queried whether a crossing, or change of the sign of the crossing function, z ( 1,0, and 1 for below, at, and above the threshold, respectively), takes place, indicated bysetting the inquiry boolean variable cross to true. The first option is used to find thenew state of all controlled junctions and the second to halt continuous simulation whena discrete event is generated.The comparison can be of the types listed in Table 4 and results in a boolean output(true and false) that can be connected to transitions between states. Several signal portsmay be connected to one transition to form a logical conjunction. A signal port withoutput true generates a discrete event that may enable a transition and when it does,force it to occur immediately (i.e., the FSM implements ‘must fire’ semantics).3.2Hybrid Bond Graph SimulationSimulation of hybrid systems has to deal with a number of idiosynchracies [6].3.2.1Event Detection and LocationWhen continuous variables exceed threshold values, as specified in the signal ports,HyBrSim uses a bisectional search to find when the first event (in general there are12
Table 4: Crossing variable, x, and its threshold, δ, comparisons.requestx δx δx δx δx δ1return valuex δ ( cross z 1)x δ ( cross z 0)(cross z 0 x δ ) (cross z 0 x δ ) ( cross x δ )x δ ( cross z 0)x δ ( cross z 1)1in0.8reliefvp0.60.40.20.2000.05relief0.60.4 0.20in0.80.10.150.20.25vp 0.20.1950.30.20.2050.21timetime(b) Zoomed in on switching behavior.(a) Aborted control maneuver.Figure 7: Simulation of the model in Fig. 6.multiple events) in the T interval occurs. If an event is detected, the stepsize is reducedfrom T to δtm , where δtm is computed based on whether an event occurs, σ 1, ornot, σ 0, in the interval δti as followsδti 1 δti ti (1 σ) ti 1 21 ti(3)The initial values for this iteration are δt0 0 and t0 T , and the iterationterminates after a fixed number of a priori prescribed steps, m. This method is robustand guaranteed to find the first event with a pre-specified accuracy, provided the crossingfunction does not have an even number of roots on the δti intervals. Because HyBrSimrequires the user to determine the step-size of the Euler method, T , this is the user’sresponsibility.3.2.2ReinitializationDuring event iteration, algebraic dependencies of storage elements may arise that discontinuously change the state values using the iteration procedure in Section 2.2.2.13
3.2.3Event IterationWhen an event occurs and a FSM changes its state, a further transition from this newstate may be immediately enabled, requiring a new evaluation of the FSM and causinga discrete iteration phase. Furthermore, in case of dynamic coupling, when a discretestate change causes a junction to switch between its on and off state (not each statechange is necessarily one between on and off), effort and flow variables in the bondgraph may change their values and when these values are propagated into the signalports they may enable further state transitions. Therefore, before continuous simulationcan resume, iteration between the discrete model parts and between the discrete andcontinuous model parts is necessary to first find a consistent, stable, continuous anddiscrete state.A Priori and a Posteriori Values Algebraic constraints may be activated and deactivated during one sequence of discrete state changes and change the state values ofstorage elements. In the bond graph model it may or may not be desired to return tothe original values before the discrete state changes [5]. A check box specifies whethera signal port generates events based on a priori or a posteriori values of x around adiscontinuity, shown by a ‘-’ or ‘ ’ sign, respectively, on the left in the signal port (seethe relief FSM in Fig. 6). In case of a posteriori conditions, the new model variablevalues computed as described in Section 2.2.13 are adopted by the signal port. In caseof a priori switching conditions, the values are only adopted after the system state isupdated and so discontinuous changes effected. The application of this is illustratednext.Mythical Modes Consider the situation where the intake valve closes while vp 0,as discussed previously. The oil viscosity, Roil , immediately causes a large pressure inthe cylinder chamber and this may cause the relief valve to open without any noticeablechange in piston velocity. If the oil viscosity, Roil and elasticity, Coil , are abstractedaway (simply removed from the bond graph in Fig. 6), the piston velocity becomes 0 inthe mode where both valves are closed. If the required (now infinitely) large pressurecauses the relief valve to open, the velocity would remain 0, which differs from behaviorof the more detailed model. Instead, the state values before the sequence of switchesstarted by closing the intake valve should be transferred to the mode where the reliefvalve is opened. The intermediate mode where both valves are closed is called a mythical3If two consecutive evaluations occur at the same point in time, i.e., tk tk 1 , the differenceapproximation to compute efforts and flows of elements in derivative causality is formed by using anξ T .14
Figure 8: Actuator with explicit state jump.mode [5, 21]. In this example, if the opening of the relief valve is based on a posteriorivalues, the state vector is not updated yet when it is inferred that the valve opens, and,therefore, the state is transferred correctly.Pinnacles When the intake valve closes, the viscous pressure may not suffice to openthe relief valve. Instead, the elasticity may further build up pressure and there is asignificant change in elevator velocity before the relief valve opens, see Fig. 7(b). If theoil parameters are abstracted away, a coefficient of restitution, , is used that dependson the dissipation of the original parameters, to compute the change in elevator velocity,vp vp , where vp is the value immediately before switching started. In Fig. 8 this isimplemented by the Sf element with 0.6.4The algebraic equation is activated using a posteriori values and deactivated using apriori values. The relief valve is opened based on a priori values to ensure the computedvelocity change is effected. The stuttering behavior shown in Fig. 7 now occurs instantaneously at the same point in time, because the oil parameters are not present anymore.A sequence of activations of the algebraic restitution constraint and opening of the reliefvalve reduces piston velocity before it can be safely set to 0, shown in Fig. 9 by the datapoints with decreasing velocity at the switching time. Note that the mode where bothvalves are closed takes a mythical incarnation, otherwise, the velocity transferred to the4Note the unique labels of connections and that causal conflicts between sources and junctions thatare off are not terminal.15
mode where pinnacle is on is 0, and no stuttering behavior takes place.Also note that in Fig. 9(b) the data points are evenly spaced during continuousintegration (distance T
The bond graph elements are rectangles with the element type on the left and their name on the right [10]. Power bonds (depicted by a harpoon) connect elements that exchange energy, e.g., in Fig. 2 the connection between the Se element and the TF element. In addition,
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